tailieunhanh - Đề tài " Kloosterman identities over a quadratic extension "

We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables. 1. Introduction One of the simplest examples of Langlands’ principle of functoriality is the quadratic base change. Namely, let E/F be a quadratic extension of global fields and z → z the corresponding Galois conjugation. The base change associates to every automorphic representation π of GL(n, F) an automorphic representation Π of GL(n,E). If n = 1 then π is an id`ele class character and Π(z) = π(zz) | Annals of Mathematics Kloosterman identities over a quadratic extension By Herv re Jacquet Annals of Mathematics 160 2004 755 779 Kloosterman identities over a quadratic extension By Hervé Jacquet Contents 1. Introduction 2. Proof of Proposition 1 3. The Kloosterman transform 4. Key lemmas 5. Proof of Proposition 3 6. Complement Abstract We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables. 1. Introduction One of the simplest examples of Langlands principle of functoriality is the quadratic base change. Namely let E F be a quadratic extension of global fields and z z the corresponding Galois conjugation. The base change associates to every automorphic representation n of GL n F an automorphic representation n of GL n E . If n 1 then n is an idele class character and n z n zz . An automorphic representation n of GL n E is a base change if and only if it is invariant under the Galois action. The existence of the base change is established by the twisted trace formula 3 . Formally if f and f are smooth functions of compact support on G EA and G FA respectively then one defines Kf x y 52 f x-1 Kf x y 52 f x i y The author was partially supported by NSF grant DMS-9619766. 756 HERVE JACQUET The identity of the twisted trace formula is that ịKf x x dx 1K . for many pairs of functions f f . The existence of such an identity depends on a simple relation between orbital integrals of the form y f xYx-i dx Ị f xY x 1 dx. In turn to establish such a relation one needs to compare at almost all places v of F inert in E the orbital integrals of specific functions. This is the fundamental lemma 9 . There is another possible characterization of the base change. Indeed in the case n 1 n is a base change if and only if it is trivial on the group of elements of norm 1 that is on the unitary group in one variable. Thus it is natural to conjecture that a representation n is a base change if .

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